Sub-sampling and parametric estimation for multiscale dynamics

Abstract

We study the problem of adequate data sub-sampling for consistent parametric estimation of unobservable stochastic differential equations (SDEs), when the data are generated by multiscale dynamic systems approximating these SDEs in some suitable sense. The challenge is that the approximation accuracy is scale dependent, and degrades at very small temporal scales. Therefore, maximum likelihood parametric estimation yields inconsistent results when the sub-sampling time-step is too small. We use data from three multiscale dynamic systems-the Additive Triad, the Truncated Burgers-Hopf models, and the Model with the Fast-Oscillating Potential-to illustrate this sub-sampling problem. In addition, we also discuss an important practical question of constructing the bias-corrected estimators for a fixed but unknown value of the multiscale parameter.